Atomic Nuclear Physics

Aharonov-Bohm and other Cyclic Phenomena by James Hamilton

By James Hamilton

The Aharonov-Bohm impression is linked to cyclic movement. it's one among a couple of anholonomic results, and which means the dynamical description is determined by the present place of the procedure and at the direction wherein it reached that place. An instance of an anholonomic impact is Foucault's recognized pendulum, which easily demonstrates the Earth's rotation. The Sagnac influence - a mild beam passing round a circled procedure of mirrors - is one other instance. glossy dynamical advancements similar to Hannay's attitude and Berry's part are extra invaluable examples.

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Example text

Consider an infinite cylindrical solenoid as in Fig. 23]. In order to see what is wrong with this argument, it is useful to remember the basic idea of the gauge transformation in the case of wave mechanics. 10a) Thus the electron probability destribution I~b(x)l2 is unaltered by the gauge transformation. 13a). 11a) to the eigenvalue of Lz. 13), as the total component of the electron's angular momentum about the symmetry axis Oz. 3 Conclusion The argument in Refs. 11), that - on applying gauge invariance consistently - leads to the operator Az which 56 5.

A field defined in this way will guide the electron, of speed v, around the closed loop. Thus non-circular orbits do exist in axially symmetric fields of the form B z ( p ) , so long as B z is not uniform. 4) K is a constant along the electron's path, and that path need not be a circle, or even a closed loop. The value of K is arbitrary. We can write A - epAo = K , A being the (ordinary) angular momentum about Oz. 4a) is called the law of conservation of canonical angular momentum. The relation shows very neatly how A is forced to vary by the non-central nature of the Lorentz force.

W e choose a point O on the circle. ) 0 being the angle measured from O. ) . In the first case the difference in phase factor on going once around the circle from O to O is obtained by putting 0 --* 2~r, so we have exp(21rik) in t h a t case. In the second case 0 --, - 2 ~ is used in going from O to O, again giving exp(21rik) . This can give constructive interference at O and that in turn can give rise to a large probability density Ir there. This will be related to the presence or absence of resistance.

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